What impact did fiscal policy actions taken during the Great Recession (2007-09) have on U.S. economic performance?

Bush passed the Economic Stimulus Act of 2008 and also the Emergency Economic Stabilization Act of 2008 which established the Troubled Assets Relief Program. Obama passed the American Recovery and Reinvestment Act in early 2009.
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1 Answer
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The fiscal policies to fight the great recession in the United States were not administered in a way that would allow us to assess this a natural experiment. However, there have been many situations in the past where there was such a natural-experiment setting, and that allows us to approximate the effect of these fiscal policies. For example, if a dollar of additional government spending, when such variation is experimental in nature creates two dollars in private consumption and investment (a multiplier of two), then when the government spends a trillion dollars in a way that may be endogenous, you could guess that the effect on private consumption would be something like two trillion.

The American Recovery and Reinvestment Act of 2009 (ARRA) contains
provisions that are intended to boost economic activity and employment
in the United States. Section 1512(e) of the law requires the
Congressional Budget Office (CBO) to comment on reports filed by
recipients of ARRA funding that detail the number of jobs funded
through their activities. This CBO report fulfills that requirement.
It also provides CBO's estimates of ARRA's overall impact on
employment and economic output in the fourth quarter of calendar year
2010. Those estimates—which CBO considers more comprehensive than the recipients' reports—are based on evidence from similar policies
enacted in the past and on the results of various economic models.

Key findings:

On that basis, CBO estimates that ARRA’s policies had the following
effects in the fourth quarter of calendar year 2010:

Lowered the unemployment rate by between 0.7 percentage points and
1.9 percentage points, Increased the number of people employed by between 1.3 million and 3.5 million, and

Increased the number of full-time-equivalent jobs by 1.8 million to
5.0 million compared with what would have occurred otherwise, as shown in Table 1. (Increases in FTE jobs include shifts from
part-time to full-time work or overtime and are thus generally
larger than increases in the number of employed workers.)

The recession from 2007 to 2009 sparked wide interest in the economic
effects of fiscal policy. That interest is reflected in an ongoing
debate over the size of the fiscal multiplier. This working paper
addresses three questions: What models do economists use to estimate
that multiplier? Why do estimates of it vary widely? How can
economists use those estimates to judiciously analyze U.S. economic
policy?

Tables 1 and 2 are of particular interest, they can help you understand the estimated contributions of various parts of the stimulus to the recovery by time and project type.